Remarkable Lives an Legacy of Sofia Kovalevskaya and Emmy Noether

Remarkable lives an legacy of Soﬁa Kovalevskaya and Emmy Noether

Leon A. Takhtajan

Stony Brook University, Stony Brook, USA Euler Mathematical Institute, Saint Petersburg, Russia

ICTS Public Lecture Bangalore, January 10, 2017 2 Amalie Emmy Noether In Erlangen and Göttingen Hilbert’s assistant Noether theorem Abstract algebra Tribute

Plan

1 Soﬁa Vasilyevna Kovalevskaya Early years Higher mathematics Major work Plan

1 Soﬁa Vasilyevna Kovalevskaya Early years Higher mathematics Major work

2 Amalie Emmy Noether In Erlangen and Göttingen Hilbert’s assistant Noether theorem Abstract algebra Tribute Soﬁa Vasilyevna Kovalevskaya

Soﬁa Kovalevskaya, 1850–1891

Soﬁa Kovalevskaya (née Korvin-Krukovskaya), was born in Moscow on 15 January 1850. Her father, lieutenant general Vasily Vasilyevich Korvin-Krukovsky, was a head of the Moscow artillery and her mother, Yelizaveta Fedorovna Schubert, was from a family of German scholars who had settled in Saint Petersburg during the time of Catherine the Great. Soﬁa’s maternal grandfather was general Theodor Friedrich von Schubert, a head of the Russian military topographic service.

Theodor Friedrich von Schubert, 1789–1865

Soﬁa had a typical upbringing for a girl of her class and time. She was left largely in the care of nurses and governesses, spoke English and French almost as well as she did Russian, and was reared in the belief that her future would be settled by the marriage with a young man of suitable wealth and family position. • The wall of Soﬁa’s room in the family country estate in lieu of the wallpaper were papered with pages by the the lecture notes by Ostrogradsky on differential and integral calculus. • Gauss-Ostrogradsky theorem Ñ Ó ( F)dV (F n)dS V ∇ · = S ·

Soﬁa introduction to mathematics • Soﬁa’s father had a love of science and mathematics and in his student days attended lectures by M.V. Ostrogradsky

Mikhail Vasilyevich Ostrogradsky, 1801–1862 • Gauss-Ostrogradsky theorem Ñ Ó ( F)dV (F n)dS V ∇ · = S ·

Soﬁa introduction to mathematics • Soﬁa’s father had a love of science and mathematics and in his student days attended lectures by M.V. Ostrogradsky

Mikhail Vasilyevich Ostrogradsky, 1801–1862

• The wall of Sofia’s room in the family country estate in lieu of the wallpaper were papered with pages by the the lecture notes by Ostrogradsky on differential and integral calculus. Sofia introduction to mathematics • Sofia’s father had a love of science and mathematics and in his student days attended lectures by M.V. Ostrogradsky

Mikhail Vasilyevich Ostrogradsky, 1801–1862

• The wall of Sofia’s room in the family country estate in lieu of the wallpaper were papered with pages by the the lecture notes by Ostrogradsky on differential and integral calculus. • Gauss-Ostrogradsky theorem Ñ Ó ( F)dV (F n)dS V ∇ · = S · Childhood memories “As I speak of these, my first contacts with mathematics, I cannot help mentioning a curious circumstance which also helped to arouse my interest in the field. When we moved permanently to the country, the whole house had to be redecorated and all the rooms had to be freshly wallpapered. But since there were many rooms, there wasn’t enough wallpaper for one of the nursery rooms... But by happy chance, the paper for this preparatory covering consisted of the lithographed lectures of Professor Ostrogradsky on differential and integral calculus, which my father had acquired as young man. These sheets, all speckled over with strange, unintelligible formulas, soon attracted my attention; I remember as a child standing for hours on end in front of this mysterious wall, trying to figure out at least some isolated sentences and to find the sequence in which the sheets should follow one another. From this protracted daily contemplation, the outer appearance of many of these formulas imprinted themselves in my memory; indeed, their very text left a deep trace in my brain, although they were incomprehensible to me while I was reading them.” • In 1860th higher equation for women in Russia was not available and one needed to study abroad. Only in 1878 Women’s Higher Education Institution (Bestuzhev Courses) were opened in Saint Petersburg with lectures by famous professors Borodin, Mendeleyev, Sechenov and Zelinski. • In order to study abroad, Sofia needed written permission from her father (or husband). Accordingly, she contracted a ‘fictitious marriage’ with Vladimir Kovalevsky in 1868.

First mathematics lessons • “I took my first lesson in differential calculus from the eminent Petersburg Professor Aleksandr Nikolaevich Strannolyubsky. He was amazed at the speed with which I grasped and assimilated the concepts of limit and of derivatives, exactly as if you knew them in advance. I recall that he expressed himself in just those words. And, as a matter of fact, at the moment when he was explaining these concepts I suddenly had a vivid memory of all this, written on the memorable sheets of Ostrogradsky; and the concept of limit appeared to me as an old friend.” • In 1860th higher equation for women in Russia was not available and one needed to study abroad. Only in 1878 Women’s Higher Education Institution (Bestuzhev Courses) were opened in Saint Petersburg with lectures by famous professors Borodin, Mendeleyev, Sechenov and Zelinski. First mathematics lessons • “I took my first lesson in differential calculus from the eminent Petersburg Professor Aleksandr Nikolaevich Strannolyubsky. He was amazed at the speed with which I grasped and assimilated the concepts of limit and of derivatives, exactly as if you knew them in advance. I recall that he expressed himself in just those words. And, as a matter of fact, at the moment when he was explaining these concepts I suddenly had a vivid memory of all this, written on the memorable sheets of Ostrogradsky; and the concept of limit appeared to me as an old friend.” • In 1860th higher equation for women in Russia was not available and one needed to study abroad. Only in 1878 Women’s Higher Education Institution (Bestuzhev Courses) were opened in Saint Petersburg with lectures by famous professors Borodin, Mendeleyev, Sechenov and Zelinski. • In order to study abroad, Sofia needed written permission from her father (or husband). Accordingly, she contracted a ‘fictitious marriage’ with Vladimir Kovalevsky in 1868. Vladimir Onufrievich Kovalevsky, 1842–1883

• Vladimir Kovalevsky was a Russian revolutioner (belonged to the Russian nihilist movement’, a kind of utopian socialism), geologist, paleontologist, founder of evolution paleonthology. • Same year — went to London with her husband, met Thomas Huxley and Charles Darwin and debated with Herbert Spencer “woman’s capacity for abstract thought”.

Herbert Spencer, 1820–1903

Higher mathematics: Heidelberg and Berlin • 1869 — Kovalevskaya attended the University of Heidelberg, Germany (was allowed to audit classes with professors approval). • 1869-1871 — Kovalevskaya attended lectures by Hermann von Helmholtz and Gustav Kirchhoff. • 1871 — took private lessons with Karl Weierstrass in Berlin (the university did not allow to audit classes).

Higher mathematics: Heidelberg and Berlin • 1869 — Kovalevskaya attended the University of Heidelberg, Germany (was allowed to audit classes with professors approval). • Same year — went to London with her husband, met Thomas Huxley and Charles Darwin and debated with Herbert Spencer “woman’s capacity for abstract thought”.

Herbert Spencer, 1820–1903 • 1871 — took private lessons with Karl Weierstrass in Berlin (the university did not allow to audit classes).

Herbert Spencer, 1820–1903

• 1869-1871 — Kovalevskaya attended lectures by Hermann von Helmholtz and Gustav Kirchhoff. Higher mathematics: Heidelberg and Berlin • 1869 — Kovalevskaya attended the University of Heidelberg, Germany (was allowed to audit classes with professors approval). • Same year — went to London with her husband, met Thomas Huxley and Charles Darwin and debated with Herbert Spencer “woman’s capacity for abstract thought”.

Herbert Spencer, 1820–1903

• 1869-1871 — Kovalevskaya attended lectures by Hermann von Helmholtz and Gustav Kirchhoff. • 1871 — took private lessons with Karl Weierstrass in Berlin (the university did not allow to audit classes). Hermann Ludwig Ferdinand Gustav Robert Kirchhoff, von Helmholtz, 1821–1894 1824–1887 Karl Theodor Wilhelm Weierstrass, 1815–1897

• Called Soﬁa the most talented of his students (who included such eminent mathematicians as Georg Frobenius, Hermann Schwarz and Carl Runge) and watched out for her interests as carefully as he did for his own. • Paper on Laplace’s calculations on the shape of Saturn rings. • Paper on the reduction of abelian integrals to elliptic integrals. • Paper on local existence and uniqueness theorem for Cauchy initial value problem for partial differential equations with analytic coefﬁcients — the celebrated Cauchy-Kovalevskaya theorem. • Unknown to Weierstrass and Kovalevskaya, Cauchy proved a special case of this theorem in 1842. Starting with Poincaré, Hermite and others it was acknowledged that Kovalevskaya’s elegant method proves the general case. She also noted that certain equations have no solutions even when they have “formal power series“” solutions, famous Kovalevskaya example.

Doctoral dissertation based on three papers

• Was granted doctoral degree summa cum laude from Göttingen University (in absentia) in 1874. Sofia Kovalevskaya was the first woman to be granted a doctoral degree in mathematics. • Paper on the reduction of abelian integrals to elliptic integrals. • Paper on local existence and uniqueness theorem for Cauchy initial value problem for partial differential equations with analytic coefficients — the celebrated Cauchy-Kovalevskaya theorem. • Unknown to Weierstrass and Kovalevskaya, Cauchy proved a special case of this theorem in 1842. Starting with Poincaré, Hermite and others it was acknowledged that Kovalevskaya’s elegant method proves the general case. She also noted that certain equations have no solutions even when they have “formal power series“” solutions, famous Kovalevskaya example.

Doctoral dissertation based on three papers

• Was granted doctoral degree summa cum laude from Göttingen University (in absentia) in 1874. Sofia Kovalevskaya was the first woman to be granted a doctoral degree in mathematics. • Paper on Laplace’s calculations on the shape of Saturn rings. • Paper on local existence and uniqueness theorem for Cauchy initial value problem for partial differential equations with analytic coefficients — the celebrated Cauchy-Kovalevskaya theorem. • Unknown to Weierstrass and Kovalevskaya, Cauchy proved a special case of this theorem in 1842. Starting with Poincaré, Hermite and others it was acknowledged that Kovalevskaya’s elegant method proves the general case. She also noted that certain equations have no solutions even when they have “formal power series“” solutions, famous Kovalevskaya example.

Doctoral dissertation based on three papers

• Was granted doctoral degree summa cum laude from Göttingen University (in absentia) in 1874. Soﬁa Kovalevskaya was the ﬁrst woman to be granted a doctoral degree in mathematics. • Paper on Laplace’s calculations on the shape of Saturn rings. • Paper on the reduction of abelian integrals to elliptic integrals. • Unknown to Weierstrass and Kovalevskaya, Cauchy proved a special case of this theorem in 1842. Starting with Poincaré, Hermite and others it was acknowledged that Kovalevskaya’s elegant method proves the general case. She also noted that certain equations have no solutions even when they have “formal power series“” solutions, famous Kovalevskaya example.

Doctoral dissertation based on three papers

• Was granted doctoral degree summa cum laude from Göttingen University (in absentia) in 1874. Sofia Kovalevskaya was the first woman to be granted a doctoral degree in mathematics. • Paper on Laplace’s calculations on the shape of Saturn rings. • Paper on the reduction of abelian integrals to elliptic integrals. • Paper on local existence and uniqueness theorem for Cauchy initial value problem for partial differential equations with analytic coefficients — the celebrated Cauchy-Kovalevskaya theorem. • Unknown to Weierstrass and Kovalevskaya, Cauchy proved a special case of this theorem in 1842. Starting with Poincaré, Hermite and others it was acknowledged that Kovalevskaya’s elegant method proves the general case. She also noted that certain equations have no solutions even when they have “formal power series“” solutions, famous Kovalevskaya example. Theorem (Cauchy-Kovalevskaya) Consider the initial value problem

j ∂mu G(t,x,∂ ∂αu), 0 j m 1; j α m t = t x ≤ ≤ − + | | ≤ i ∂ u(0,x) gj(x), 0 j m 1. t = ≤ ≤ − d Suppose that gj are real analytic on a neighborhood of x0 R , and G ∈ is real analytic on a neighborhood of

j α (0,x0,∂ ∂ gj(x0), 0 j m 1; j α m). t x ≤ ≤ − + | | ≤ Then there is a real analytic solution deﬁned on a neighborhood of d (0,x0) R R . The solution is unique in the sense that if u and v are ∈ × both real analytic solutions to the equation on a connected neighborhood of (0,x0), then u v. = • In Western Europe, Kovalevskaya’s status as Weierstrass’s student would have been to her advantage, but her reputation as a nihilist was shocking to European academics, and being a woman was also an obstacle. • The most immediate barrier to her mathematical career, however, was the social convention of the time. • Kovalevskaya was married, and married women did not live apart from their husbands, nor did they support themselves with teaching positions. • Even Weierstrass thought Kovalevskaya was doing mathematics for the intellectual satisfaction. Her husband would support her and she does not need ofﬁcial recognition of her mathematical accomplishments.

Job candidate

• In Russia, the combination of being a woman and of her political views made Soﬁa unacceptable as a job candidate. • The most immediate barrier to her mathematical career, however, was the social convention of the time. • Kovalevskaya was married, and married women did not live apart from their husbands, nor did they support themselves with teaching positions. • Even Weierstrass thought Kovalevskaya was doing mathematics for the intellectual satisfaction. Her husband would support her and she does not need ofﬁcial recognition of her mathematical accomplishments.

Job candidate

• In Russia, the combination of being a woman and of her political views made Sofia unacceptable as a job candidate. • In Western Europe, Kovalevskaya’s status as Weierstrass’s student would have been to her advantage, but her reputation as a nihilist was shocking to European academics, and being a woman was also an obstacle. • The most immediate barrier to her mathematical career, however, was the social convention of the time. • Kovalevskaya was married, and married women did not live apart from their husbands, nor did they support themselves with teaching positions. • Even Weierstrass thought Kovalevskaya was doing mathematics for the intellectual satisfaction. Her husband would support her and she does not need official recognition of her mathematical accomplishments. • Weierstrass persuaded Göttingen to grant her a degree. And only after Vladimir Kovalevsky committed suicide in 1883 did Weierstrass started actively seeking a university position for Kovalevskaya. • In the meantime there was a hiatus in Kovalevskaya’s scientific activity. From 1874, when she returned to Russia, to 1878, while she was pregnant with her only daughter, Kovalevskaya more or less abandoned any attempt at original work in mathematics. • In order to teach at the higher levels, one needed a Russian master’s degree. But women were forbidden to take the exam to obtain the degree. • She plunged into literary circles, tried her hand at writing, was active in the movement to establish a women’s university in St. Petersburg, and so on.

• Only after Kovalevskaya told Weierstrass the true story of her marriage and something of her political beliefs, did he agree that a degree might be useful to her sometime in the future. • In the meantime there was a hiatus in Kovalevskaya’s scientiﬁc activity. From 1874, when she returned to Russia, to 1878, while she was pregnant with her only daughter, Kovalevskaya more or less abandoned any attempt at original work in mathematics. • In order to teach at the higher levels, one needed a Russian master’s degree. But women were forbidden to take the exam to obtain the degree. • She plunged into literary circles, tried her hand at writing, was active in the movement to establish a women’s university in St. Petersburg, and so on.

• Only after Kovalevskaya told Weierstrass the true story of her marriage and something of her political beliefs, did he agree that a degree might be useful to her sometime in the future. • Weierstrass persuaded Göttingen to grant her a degree. And only after Vladimir Kovalevsky committed suicide in 1883 did Weierstrass started actively seeking a university position for Kovalevskaya. • In the meantime there was a hiatus in Kovalevskaya’s scientiﬁc activity. From 1874, when she returned to Russia, to 1878, while she was pregnant with her only daughter, Kovalevskaya more or less abandoned any attempt at original work in mathematics. • In order to teach at the higher levels, one needed a Russian master’s degree. But women were forbidden to take the exam to obtain the degree. • Only after Kovalevskaya told Weierstrass the true story of her marriage and something of her political beliefs, did he agree that a degree might be useful to her sometime in the future. • Weierstrass persuaded Göttingen to grant her a degree. And only after Vladimir Kovalevsky committed suicide in 1883 did Weierstrass started actively seeking a university position for Kovalevskaya. • In the meantime there was a hiatus in Kovalevskaya’s scientiﬁc activity. From 1874, when she returned to Russia, to 1878, while she was pregnant with her only daughter, Kovalevskaya more or less abandoned any attempt at original work in mathematics. • In order to teach at the higher levels, one needed a Russian master’s degree. But women were forbidden to take the exam to obtain the degree. • She plunged into literary circles, tried her hand at writing, was active in the movement to establish a women’s university in St. Petersburg, and so on. Back to mathematics

Gösta Mittag-Leffler, 1846–1927 • In 1884 she became an editor of Acta Mathematica. • Appointed an extraordinary professor for five years in 1884. • Theory of partial differential equations (Fall 1884). • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves defined by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • * — following Weierstrass.

• Kovalevskaya has been offered a position of privat docent at Stockholm University in 1883. • Appointed an extraordinary professor for ﬁve years in 1884. • Theory of partial differential equations (Fall 1884). • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves deﬁned by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • * — following Weierstrass.

• Kovalevskaya has been offered a position of privat docent at Stockholm University in 1883. • In 1884 she became an editor of Acta Mathematica. • Theory of partial differential equations (Fall 1884). • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves deﬁned by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • * — following Weierstrass.

• Kovalevskaya has been offered a position of privat docent at Stockholm University in 1883. • In 1884 she became an editor of Acta Mathematica. • Appointed an extraordinary professor for ﬁve years in 1884. • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves deﬁned by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • * — following Weierstrass.

• Kovalevskaya has been offered a position of privat docent at Stockholm University in 1883. • In 1884 she became an editor of Acta Mathematica. • Appointed an extraordinary professor for five years in 1884. • Theory of partial differential equations (Fall 1884). • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves defined by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • Kovalevskaya has been offered a position of privat docent at Stockholm University in 1883. • In 1884 she became an editor of Acta Mathematica. • Appointed an extraordinary professor for five years in 1884. • Theory of partial differential equations (Fall 1884). • Theory of algebraic functions* (Spring 1885). • Elementary algebra (Spring 1885). • Theory of Abelian functions* (Fall 1885 to Spring 1887). • Theory of potential functions (Spring 1886). • Theory of motion of a rigid body (Fall 1880 and Spring 1887). • On the curves defined by differential equations, following Poincaré (Autumn 1887 and Spring 1888). • Theory of theta-functions* (Spring 1888). • Applications of the theory of elliptic functions (Autumn 1888). • Theory of partial differential equations (Spring 1890). • Theory of elliptic functions* (Autumn 1889). • Application of analysis of integer theory (Autumn 1890). • * — following Weierstrass. Paris, December 18, 1888

Permanent secretaries of the Academy to Madam Sofya Kovalevskaya in Stockholm

Madam,

We have the honour to notify You that the Academy of Sciences awarded You the Prix Bordin (improvement in an important point of the theory of motion of a solid body). We invite You, Madam, to be present at a public session that will be held on Monday, December 24 of this year at one o’clock p.m. exactly and at which time the results of the competition will be announced in public. We take this opportunity to convey our personal congratulations and testify to our confidence of the benefit the Academy anticipates Your work and Your advances will bring. Madam, receive our assurances in our highest respect. L. Pasteur J. Bertrand • Submitted anonymously under the maxim: Say what you know Do what you must Whatever will be, will be • Three integrable cases of the rotation of the heavy rigid body about the fixed point. • Euler-Poinsot case (elliptic theta-functions). • Lagrange-Poisson case (elliptic theta-functions). • Kovalevskaya case (genus 2 theta-functions). • In 1989 Kovalevskaya was appointed the chair of analysis at Stockholm University and was given a lifetime professorship — the first woman in modern times to be so honored. • Died in Stockholm in 1891 of pneumonia.

• “Sur le probleme de la rotation d’un corps solide aulour d’un point fixe“.” Acta Math., 1889, t. 12, pp. 177-232 • Three integrable cases of the rotation of the heavy rigid body about the fixed point. • Euler-Poinsot case (elliptic theta-functions). • Lagrange-Poisson case (elliptic theta-functions). • Kovalevskaya case (genus 2 theta-functions). • In 1989 Kovalevskaya was appointed the chair of analysis at Stockholm University and was given a lifetime professorship — the first woman in modern times to be so honored. • Died in Stockholm in 1891 of pneumonia.

• “Sur le probleme de la rotation d’un corps solide aulour d’un point ﬁxe“.” Acta Math., 1889, t. 12, pp. 177-232 • Submitted anonymously under the maxim: Say what you know Do what you must Whatever will be, will be • Euler-Poinsot case (elliptic theta-functions). • Lagrange-Poisson case (elliptic theta-functions). • Kovalevskaya case (genus 2 theta-functions). • In 1989 Kovalevskaya was appointed the chair of analysis at Stockholm University and was given a lifetime professorship — the ﬁrst woman in modern times to be so honored. • Died in Stockholm in 1891 of pneumonia.

• “Sur le probleme de la rotation d’un corps solide aulour d’un point fixe“.” Acta Math., 1889, t. 12, pp. 177-232 • Submitted anonymously under the maxim: Say what you know Do what you must Whatever will be, will be • Three integrable cases of the rotation of the heavy rigid body about the fixed point. • Euler-Poinsot case (elliptic theta-functions). • Lagrange-Poisson case (elliptic theta-functions). • Kovalevskaya case (genus 2 theta-functions). • In 1989 Kovalevskaya was appointed the chair of analysis at Stockholm University and was given a lifetime professorship — the first woman in modern times to be so honored. • “Sur le probleme de la rotation d’un corps solide aulour d’un point fixe“.” Acta Math., 1889, t. 12, pp. 177-232 • Submitted anonymously under the maxim: Say what you know Do what you must Whatever will be, will be • Three integrable cases of the rotation of the heavy rigid body about the fixed point. • Euler-Poinsot case (elliptic theta-functions). • Lagrange-Poisson case (elliptic theta-functions). • Kovalevskaya case (genus 2 theta-functions). • In 1989 Kovalevskaya was appointed the chair of analysis at Stockholm University and was given a lifetime professorship — the first woman in modern times to be so honored. • Died in Stockholm in 1891 of pneumonia. • Here the moments of inertia are I1 I2 2I3 where I3 1, the = = = center of mass is (x0,0,0), c Mgx0, p,q,r are components of the = angular velocitiy and γ,γ0,γ00 are cosines of angles between the z-axis of fixed coordinate system and axes of coordinate system that is attached to the rigid body and whose origin coincides with the fixed point. • Kovalevskaya solution uses theta-functions for the genus 2 hyperelliptic curve. • Solutions are meromorphic functions of complexified time t C. ∈

Kovalevskaya case

• Differential equations

2p˙ qr, γ˙ rγ0 qγ00, = = − 2q˙ pr cγ00, γ˙0 pγ00 rγ, = − − = − r˙ cγ0, γ˙00 qγ pγ0. = = − • Kovalevskaya solution uses theta-functions for the genus 2 hyperelliptic curve. • Solutions are meromorphic functions of complexiﬁed time t C. ∈

Kovalevskaya case

• Differential equations

2p˙ qr, γ˙ rγ0 qγ00, = = − 2q˙ pr cγ00, γ˙0 pγ00 rγ, = − − = − r˙ cγ0, γ˙00 qγ pγ0. = = −

• Here the moments of inertia are I1 I2 2I3 where I3 1, the = = = center of mass is (x0,0,0), c Mgx0, p,q,r are components of the = angular velocitiy and γ,γ0,γ00 are cosines of angles between the z-axis of fixed coordinate system and axes of coordinate system that is attached to the rigid body and whose origin coincides with the fixed point. • Solutions are meromorphic functions of complexified time t C. ∈

Kovalevskaya case

• Differential equations

2p˙ qr, γ˙ rγ0 qγ00, = = − 2q˙ pr cγ00, γ˙0 pγ00 rγ, = − − = − r˙ cγ0, γ˙00 qγ pγ0. = = −

• Here the moments of inertia are I1 I2 2I3 where I3 1, the = = = center of mass is (x0,0,0), c Mgx0, p,q,r are components of the = angular velocitiy and γ,γ0,γ00 are cosines of angles between the z-axis of ﬁxed coordinate system and axes of coordinate system that is attached to the rigid body and whose origin coincides with the ﬁxed point. • Kovalevskaya solution uses theta-functions for the genus 2 hyperelliptic curve. Kovalevskaya case

• Differential equations

2p˙ qr, γ˙ rγ0 qγ00, = = − 2q˙ pr cγ00, γ˙0 pγ00 rγ, = − − = − r˙ cγ0, γ˙00 qγ pγ0. = = −

• Here the moments of inertia are I1 I2 2I3 where I3 1, the = = = center of mass is (x0,0,0), c Mgx0, p,q,r are components of the = angular velocitiy and γ,γ0,γ00 are cosines of angles between the z-axis of fixed coordinate system and axes of coordinate system that is attached to the rigid body and whose origin coincides with the fixed point. • Kovalevskaya solution uses theta-functions for the genus 2 hyperelliptic curve. • Solutions are meromorphic functions of complexified time t C. ∈ Legacy

• Along with Weierstrass, Hermite, Mittag-Leffler, Picard and Poincaré, Sofia Kovalevskaya was considered one of the best mathematical analysts in Europe. Pelageya Kochina“ Love and Mathematics: Sofia Kovalevskaya” Mir Publishers, Moscow, 1985 Amalie Emmy Noether

Amalie Emmy Noether, 1882–1935 • Her father — Max Noether — a well-known mathematician working on algebraic geometry of surfaces (Noether’s formula, the Noether inequality).

Max Noether, 1844–1921

• Her mother, Ida Amalia Kaufmann, the daughter of a wealthy Jewish merchant family.

• Amalie Emmy Noether was born 23 March 1882 in Erlangen, Bavaria, Germany. • Her mother, Ida Amalia Kaufmann, the daughter of a wealthy Jewish merchant family.

• Amalie Emmy Noether was born 23 March 1882 in Erlangen, Bavaria, Germany. • Her father — Max Noether — a well-known mathematician working on algebraic geometry of surfaces (Noether’s formula, the Noether inequality).

Max Noether, 1844–1921 • Amalie Emmy Noether was born 23 March 1882 in Erlangen, Bavaria, Germany. • Her father — Max Noether — a well-known mathematician working on algebraic geometry of surfaces (Noether’s formula, the Noether inequality).

Max Noether, 1844–1921

• Her mother, Ida Amalia Kaufmann, the daughter of a wealthy Jewish merchant family. • One of only two women of 986 students, she was only allowed to audit classes with the permission of individual professors. • During the 1903-04 winter semester, Noether studied at the University of Göttingen, attending lectures by the astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Felix Klein, and David Hilbert. • Restrictions on women’s participation were rescinded and Noether returned to the University of Erlangen in the fall 1904 to major in mathematics. • Under the supervision of Paul Gordan in 1907 she wrote her dissertation, “Über die Bildung des Formensystems der ternären biquadratischen Form” (“On complete systems of invariants for ternary biquadratic forms”). • In 1908–15 taught at the University of Erlangen’s Mathematical Institute without pay.

• In 1900 Noether started her studies at the University of Erlangen. • During the 1903-04 winter semester, Noether studied at the University of Göttingen, attending lectures by the astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Felix Klein, and David Hilbert. • Restrictions on women’s participation were rescinded and Noether returned to the University of Erlangen in the fall 1904 to major in mathematics. • Under the supervision of Paul Gordan in 1907 she wrote her dissertation, “Über die Bildung des Formensystems der ternären biquadratischen Form” (“On complete systems of invariants for ternary biquadratic forms”). • In 1908–15 taught at the University of Erlangen’s Mathematical Institute without pay.

• In 1900 Noether started her studies at the University of Erlangen. • One of only two women of 986 students, she was only allowed to audit classes with the permission of individual professors. • Restrictions on women’s participation were rescinded and Noether returned to the University of Erlangen in the fall 1904 to major in mathematics. • Under the supervision of Paul Gordan in 1907 she wrote her dissertation, “Über die Bildung des Formensystems der ternären biquadratischen Form” (“On complete systems of invariants for ternary biquadratic forms”). • In 1908–15 taught at the University of Erlangen’s Mathematical Institute without pay.

• In 1900 Noether started her studies at the University of Erlangen. • One of only two women of 986 students, she was only allowed to audit classes with the permission of individual professors. • During the 1903-04 winter semester, Noether studied at the University of Göttingen, attending lectures by the astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Felix Klein, and David Hilbert. • Restrictions on women’s participation were rescinded and Noether returned to the University of Erlangen in the fall 1904 to major in mathematics. • Under the supervision of Paul Gordan in 1907 she wrote her dissertation, “Über die Bildung des Formensystems der ternären biquadratischen Form” (“On complete systems of invariants for ternary biquadratic forms”). • In 1900 Noether started her studies at the University of Erlangen. • One of only two women of 986 students, she was only allowed to audit classes with the permission of individual professors. • During the 1903-04 winter semester, Noether studied at the University of Göttingen, attending lectures by the astronomer Karl Schwarzschild and mathematicians Hermann Minkowski, Felix Klein, and David Hilbert. • Restrictions on women’s participation were rescinded and Noether returned to the University of Erlangen in the fall 1904 to major in mathematics. • Under the supervision of Paul Gordan in 1907 she wrote her dissertation, “Über die Bildung des Formensystems der ternären biquadratischen Form” (“On complete systems of invariants for ternary biquadratic forms”). • In 1908–15 taught at the University of Erlangen’s Mathematical Institute without pay. • The philologists and historians objected: “What will our soldiers think when they return to the university and ﬁnd that they are required to learn at the feet of a woman?” • Hilbert responded with indignation, stating, “I do not see that the sex of the candidate is an argument against her admission as privat dozent. After all, we are a university, not a bath house.”

David Hilbert, 1862–1943

• In 1915, Noether was invited to the University of Göttingen by David Hilbert and Felix Klein. • Hilbert responded with indignation, stating, “I do not see that the sex of the candidate is an argument against her admission as privat dozent. After all, we are a university, not a bath house.”

David Hilbert, 1862–1943

• In 1915, Noether was invited to the University of Göttingen by David Hilbert and Felix Klein. • The philologists and historians objected: “What will our soldiers think when they return to the university and ﬁnd that they are required to learn at the feet of a woman?” • In 1915, Noether was invited to the University of Göttingen by David Hilbert and Felix Klein. • The philologists and historians objected: “What will our soldiers think when they return to the university and ﬁnd that they are required to learn at the feet of a woman?” • Hilbert responded with indignation, stating, “I do not see that the sex of the candidate is an argument against her admission as privat dozent. After all, we are a university, not a bath house.”

David Hilbert, 1862–1943 • Applications include conservation of energy, momentum and angular momentum in classical mechanics, energy-momentum tensor in electromagnetism, special and general relativity, charge conservation, etc. • “Noether theorem certainly is one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem” (in “Symmetry and the Beautiful Universe” by L.M. Lederman and C.T. Hill). • In classical physics Noether theorem uses equations of motion while its analog in quantum physics deals with expectation values of quantum ﬁelds.

Noether theorem

• In 1915 Emmy Noether proved the following theorem (published in 1918): Every differentiable symmetry of the action of a physical system has a corresponding conservation law. • “Noether theorem certainly is one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem” (in “Symmetry and the Beautiful Universe” by L.M. Lederman and C.T. Hill). • In classical physics Noether theorem uses equations of motion while its analog in quantum physics deals with expectation values of quantum ﬁelds.

Noether theorem

• In 1915 Emmy Noether proved the following theorem (published in 1918): Every differentiable symmetry of the action of a physical system has a corresponding conservation law. • Applications include conservation of energy, momentum and angular momentum in classical mechanics, energy-momentum tensor in electromagnetism, special and general relativity, charge conservation, etc. • “Noether theorem certainly is one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem” (in “Symmetry and the Beautiful Universe” by L.M. Lederman and C.T. Hill). • In classical physics Noether theorem uses equations of motion while its analog in quantum physics deals with expectation values of quantum fields. • Noetherian module is a module that satisfies the ascending chain condition on its submodules. • Noetherian scheme is a scheme that admits a finite covering by open affine subsets Spec(Ai), where Ai are Noetherian rings. • Noether normalization lemma — an important step in proving Hilbert’s Nullstellensatz. • The Lasker-Noether theorem: in a Noetherian ring every ideal can be decomposed as a product of finitely many primary ideals. • Emmy Noether “changed the face of algebra by her work” (Hermann Weyl).

Abstract algebra

• Noetherian ring — a ring that satisﬁes the ascending chain condition on ideals: for any such chain of ideals